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Circles, Distance and Midpoint Formulas Objective 1: Distance Formula

Let (x1, y1) and (x2, y2) be any two points on the

plane. Then by the Pythagorean Theorem,

( ) ( )2 22

2 1 2 1d x x y y= − + −

( ) ( )2 22

2 1 2 1d x x y y= − + −

( ) ( )2 2

2 1 2 1d x x y y= − + −

Distance Formula

The distance between two points ( )1 1,x y and ( )2 2,x y is

( ) ( )2 2

2 1 2 1d x x y y= − + −

Distance between x and y on a

number line is x y y x− = −

Example: Find the distance between the points ( )4, 5− and ( )2,3− .

Cypress College Math Department – CCMR Notes Circles, Distance and Midpoint Formulas, Page 1 of 18

Example: Prove that the three points ( )0,0 , ( )3,3 , and ( )1,1− form a right triangle.

Pause the video to try this one on your own, then restart when you are ready to check

your answer.

Extra Practice

1. Find the distance between the points ( )1, 2− and ( )4, 2 .

2. Prove that the three points ( )0,0 , ( )2, 2− − , and ( )1, 1− form a right triangle.

Restart when you are ready to check your answers.


Objective 2: Midpoint Formula

Midpoint between a and b on a number line is 2

a b+

Midpoint Formula

The midpoint ( ),M x y= of the line segment joining ( )1 1,x y and ( )2 2,x y is

1 2 1 2,2 2

x x y yM

+ + =

Example: Find the midpoint of the line segment joining ( )8, 1− and ( )6,7− .


Example: Find the midpoint of the line segment joining 3

6,2

and 5

, 124

−

.


your answer.

Extra Practice

1. Find the midpoint of the line segment joining ( )7,10− and ( )3, 2− .

2. Find the midpoint of the line segment joining 1

, 12

−

and

84,

3

−

.



Objective 3: Equations and Graphs of Circles

A circle is one of the conic sections.

A conic section is the intersection of

a plane with a right circular cone.

Definition: A circle is a set of points in a plane that are equidistant from a fixed point called the center. The fixed distance from the center is called the radius.

The center of a circle is the fixed point (h, k)

equidistant from the points (x, y) on the circle.

The radius of a circle is the fixed distance r

from the center (h, k) to the points (x, y) on the

circle.

Distance from any point on the circle to the center = r

The distance from center (h, k) to the points (x, y) on the circle must equal to r.

( ) ( )2 2

2 1 2 1d x x y y= − + −

Distance between (x, y) and (h, k) r=

( ) ( )2 2

r x h y k= − + −

By squaring both sides of the equation,

( ) ( )2 2 2x h y k r− + − =


Standard Form for the Equation of a Circle

( ) ( )2 2 2x h y k r− + − = center ( ),h k

2 2 2x y r+ = center ( )0,0

Example: Match the graph to its equation.

A. 2 2 9x y+ =

B. 2 2 3x y+ =

C. 2 2 9x y− =

D. 2 2 0x y− =


A. ( )2 21 4x y− + =

B. ( )2 1 16x y+ + =

C. ( )2 21 16x y+ + =

D. ( )22 1 4x y+ − =



A. ( ) ( )2 2

2 1 25x y+ + − =

B. ( ) ( )2 2

2 1 5x y+ + − =

C. ( ) ( )2 2

2 1 5x y− + + =

D. ( ) ( )2 2

2 1 25x y− + + =

Example: Determine the equation of the circle with center (0, -9) and radius 4

5.

Example: Determine the equation of the circle with center (-7, 6) and point on the circle (4,-2).

Example: Determine the equation of the circle with two endpoints of a diameter (-5, 6) and

(3, -8).



your answer.

Extra Practice

1. Match the graph to its equation.

A.

B.

C.

D.

2 2 2x y+ =

2 2 4x y+ =

2 2 16x y+ =

2 2 2x y+ =

2. Match the graph to its equation.

A. ( )22 2 36x y+ − =

B. ( )22 2 6x y+ + =

C. ( )2 22 36x y− + =

D. ( )2 22 6x y+ + =


3.Match the graph to its equation.

A. ( ) ( )2 2

3 1 4x y− + + =

B. ( ) ( )2 2

3 1 16x y− + + =

C. ( ) ( )2 2

3 1 4x y+ + − =

D. ( ) ( )2 2

3 1 16x y+ + − =

4. Determine the equation of the circle with center (-6, 0) and radius 2

3.

5. Determine the equation of the circle with center (9, -11) and point on the circle (2,-7).

6. Determine the equation of the circle with two endpoints of a diameter (-3, 2) and

(5, -6).



Objective 4: Convert from General Form to Standard Form

General form for the equation of a conic section:

2 2 0Ax By Cx Dy F+ + + + =

Standard form for the equation of a circle:

( ) ( )2 2 2x h y k r− + − =

Example: Convert the following equation to standard form. Identify the center and radius.

2 26 7 0x x y− + − =

Example: Convert the following equation to standard form. Identify the center and radius.

2 2 2 12 17 0x y x y+ + − + =



your answer.

Extra Practice

1. Convert the following equation to standard form. Identify the center and radius.

2 2 10 24 0x y y+ − − =

2. Convert the following equation to standard form. Identify the center and radius.

2 2 8 6 7 0x y x y+ − + + =



Objective 5: Graph Circles

When graphing a circle:

1. Find the center (h, k) and the radius r.

2. Plot the center (h, k).

3. Plot two points r units to the left and to the

right of the center and two points r units above

and below of the center.

Example: Graph 2 2 36x y+ = . Make sure to label the exact coordinates of the center and the

four points on the circle.


Example: Graph ( )2 23 16x y− + = . Make sure to label the exact coordinates of the center and

the four points on the circle.

Example: Graph ( ) ( )2 2

1 2 9x y+ + − = . Make sure to label the exact coordinates of the center

and the four points on the circle.


Example: Graph 2 2 2 6 15 0x y x y+ − + − = . Make sure to label the exact coordinates of the

center and the four points on the circle.


your answer.

Extra Practice

2 2 64x y+ =1. Graph . Make sure to label the exact coordinates of the center and the four points on the circle.


2. Graph ( )22 2 16x y+ + = . Make sure to label the exact coordinates of the center and the

four points on the circle.

3. Graph ( ) ( )2 2

4 1 16x y+ + − = . Make sure to label the exact coordinates of the center and

the four points on the circle.


4. Graph 2 2 10 6 25 0x y x y+ + − + =

center and the four points on the circle.

. Make sure to label the exact coordinates of the


Objective 6: Applications of Circles

Example: There is a giant wheel with diameter 150 feet which sits on a 12-foot tall platform making the overall height 162 feet. Find an equation for the wheel assuming that its center lies on the y-axis

and the x-axis runs along the ground.


Example: A square is inscribed in a circle with the following equation. Find the area of the

shaded region. Round your answer to one decimal place.


your answer.

Extra Practice

1. There is a giant wheel with diameter 168 feet which sits on a 15-foot-tall platform making it overall

height of 183 feet. Find an equation for the wheel assuming that its center lies on the y-axis, and the x-

axis lies along the ground.


2. A square is inscribed in a circle with the following equation. Find the area of the shaded

region.



2: 2_2: fill_1:

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